문제
하늘이는 최근 입대하였다. 하늘이의 선임은 하늘이가 사회에서 코딩하다 온 것을 알고, 하늘이에게 다음과 같이 ”최소 스패닝 트리 다시 그리기 놀이”를 알려 주었다.
- 먼저 선임이 후임에게 무향 연결 그래프 $G$를 준다. 이때, $G$를 구성하는 간선의 가중치는 모두 10ドル^6$이하의 양의 정수이다.
- 후임은 $G$에서 그릴 수 있는 서로 다른 최소 스패닝 트리 중 동일한 확률로 최소 스패닝 트리 $T$를 하나 고른다.
- 선임은 1ドル$부터 10ドル^6$까지 차례대로 각각 독립적으로, 정수 $x$를 $\frac{1}{x}$의 확률로 후임에게 말한다. 즉, 1ドル$은 반드시 말하므로, 최소 1ドル$개 이상 10ドル^6$개 이하의 수를 말한다.
- 후임은 선임이 $x$를 말하면 $T$에서 가중치가 $x$인 간선을 모두 지운다. $T$에서 간선들이 지워진 상태를 $T^\prime$이라고 하자.
- 후임은 $T^\prime$에, 원래 그래프 $G$에 있는 간선을 0ドル$개 이상 추가하여 다시 $G$의 최소 스패닝 트리를 만든다.
하늘이는 이 놀이를 듣고, 새로 그릴 수 있는 최소 스패닝 트리의 개수가 몇 개인지 궁금해졌다. 선임이 후임에게 주는 그래프 $G$가 주어졌을 때, ”최소 스패닝 트리 다시 그리기 놀이”를 통해 그릴 수 있는 최소 스패닝 트리의 개수의 기댓값을 구해보자.
출력
$G$에 대해 “최소 스패닝 트리 다시 그리기 놀이”를 통해 그릴 수 있는 최소 스패닝 트리의 개수의 기댓값을 10ドル^9+7$로 나눈 나머지를 출력한다.
기약 분수 $\frac{p}{q}(p\ge 0,q>0,\gcd(p,q) =1)$를 $M$으로 나눈 나머지는 $q^{-1}$가 $q\cdot q^{-1}\equiv 1\pmod M$을 만족하는 정수, 즉 $q$의 $M$에 대한 모듈로 곱셈 역원일 때, $p\cdot q^{-1}\pmod M$로 정의한다. 만약 정수일 경우 $q=q^{-1}=1$이므로 $p\pmod M$를 의미한다.
만약 정답의 분모가 10ドル^9+7$의 배수인 경우 기댓값 대신 -1을 출력한다.
서브태스크
| 번호 | 배점 | 제한 | | 1 | 23 | $E = V-1$
|
| 2 | 66 | $V\le 4$
|
| 3 | 65 | 모든 $w_i = 1;$ $E=\frac{V(V-1)}{2}$
|
| 4 | 49 | 모든 $w_i = 1$
|
| 5 | 95 | 모든 $w_i \le 2$
|
| 6 | 152 | 추가적인 제한 조건 없음
|
주어진 그래프는 아래 그림과 같다.
이 그래프의 최소 스패닝 트리는 다음 16ドル$가지로, $\frac{1}{16}$의 확률로 하나를 후임이 고른다.
어떤 그래프를 골라도, $T^\prime$은 반드시 아래와 같은 상태가 되고, 다시 16ドル$개의 최소 스패닝 트리를 모두 만들 수 있다.
주어진 그래프는 아래 그림과 같다.
이 그래프의 최소 스패닝 트리는 다음 2ドル$가지로, $\frac{1}{2}$의 확률로 하나를 후임이 고른다.
이후 가능한 조합은 다음과 같다. 기댓값은 $\frac{3}{2}$이다.
[{"problem_id":"31545","problem_lang":"0","title":"\ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac \ub2e4\uc2dc \uadf8\ub9ac\uae30 \ub180\uc774","description":"<p>\ud558\ub298\uc774\ub294 \ucd5c\uadfc \uc785\ub300\ud558\uc600\ub2e4. \ud558\ub298\uc774\uc758 \uc120\uc784\uc740 \ud558\ub298\uc774\uac00 \uc0ac\ud68c\uc5d0\uc11c \ucf54\ub529\ud558\ub2e4 \uc628 \uac83\uc744 \uc54c\uace0, \ud558\ub298\uc774\uc5d0\uac8c \ub2e4\uc74c\uacfc \uac19\uc774 &rdquo;\ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac \ub2e4\uc2dc \uadf8\ub9ac\uae30 \ub180\uc774&rdquo;\ub97c \uc54c\ub824 \uc8fc\uc5c8\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uba3c\uc800 \uc120\uc784\uc774 \ud6c4\uc784\uc5d0\uac8c \ubb34\ud5a5 \uc5f0\uacb0 \uadf8\ub798\ud504 $G$\ub97c \uc900\ub2e4. \uc774\ub54c, $G$\ub97c \uad6c\uc131\ud558\ub294 \uac04\uc120\uc758 \uac00\uc911\uce58\ub294 \ubaa8\ub450 $10^6$\uc774\ud558\uc758 \uc591\uc758 \uc815\uc218\uc774\ub2e4.<\/li>\r\n\t<li>\ud6c4\uc784\uc740 $G$\uc5d0\uc11c \uadf8\ub9b4 \uc218 \uc788\ub294 \uc11c\ub85c \ub2e4\ub978 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac \uc911 \ub3d9\uc77c\ud55c \ud655\ub960\ub85c \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac $T$\ub97c \ud558\ub098 \uace0\ub978\ub2e4.<\/li>\r\n\t<li>\uc120\uc784\uc740 $1$\ubd80\ud130 $10^6$\uae4c\uc9c0 \ucc28\ub840\ub300\ub85c \uac01\uac01 \ub3c5\ub9bd\uc801\uc73c\ub85c, \uc815\uc218 $x$\ub97c $\\frac{1}{x}$\uc758 \ud655\ub960\ub85c \ud6c4\uc784\uc5d0\uac8c \ub9d0\ud55c\ub2e4. \uc989, $1$\uc740 \ubc18\ub4dc\uc2dc \ub9d0\ud558\ubbc0\ub85c, \ucd5c\uc18c $1$\uac1c \uc774\uc0c1 $10^6$\uac1c \uc774\ud558\uc758 \uc218\ub97c \ub9d0\ud55c\ub2e4.<\/li>\r\n\t<li>\ud6c4\uc784\uc740 \uc120\uc784\uc774 $x$\ub97c \ub9d0\ud558\uba74 $T$\uc5d0\uc11c \uac00\uc911\uce58\uac00 $x$\uc778 \uac04\uc120\uc744 <strong>\ubaa8\ub450<\/strong> \uc9c0\uc6b4\ub2e4. $T$\uc5d0\uc11c \uac04\uc120\ub4e4\uc774 \uc9c0\uc6cc\uc9c4 \uc0c1\ud0dc\ub97c $T^\\prime$\uc774\ub77c\uace0 \ud558\uc790.<\/li>\r\n\t<li>\ud6c4\uc784\uc740 $T^\\prime$\uc5d0, \uc6d0\ub798 \uadf8\ub798\ud504 $G$\uc5d0 \uc788\ub294 \uac04\uc120\uc744 $0$\uac1c \uc774\uc0c1 \ucd94\uac00\ud558\uc5ec \ub2e4\uc2dc $G$\uc758 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\ub97c \ub9cc\ub4e0\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ud558\ub298\uc774\ub294 \uc774 \ub180\uc774\ub97c \ub4e3\uace0, \uc0c8\ub85c \uadf8\ub9b4 \uc218 \uc788\ub294 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\uac00 \uba87 \uac1c\uc778\uc9c0 \uad81\uae08\ud574\uc84c\ub2e4. \uc120\uc784\uc774 \ud6c4\uc784\uc5d0\uac8c \uc8fc\ub294 \uadf8\ub798\ud504 $G$\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, &rdquo;\ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac \ub2e4\uc2dc \uadf8\ub9ac\uae30 \ub180\uc774&rdquo;\ub97c \ud1b5\ud574 \uadf8\ub9b4 \uc218 \uc788\ub294 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\uc758 \uae30\ub313\uac12\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\r\n","input":"<p>\uccab \ubc88\uc9f8 \uc904\uc5d0 $G$\uc758 \uc815\uc810\uc758 \uac1c\uc218 $V$\uacfc \uac04\uc120\uc758 \uac1c\uc218 $E$\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(2\\le V\\le 500;$ $V-1\\le E\\le\\frac{V(V-1)}{2})$<\/p>\r\n\r\n<p>\ub450 \ubc88\uc9f8 \uc904\ubd80\ud130 $E$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0 $G$\uc758 \uac04\uc120\uc758 \uc815\ubcf4\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 $u_i$, $v_i$, $w_i$\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774\ub294 $u_i$\ubc88 \uc815\uc810\uacfc $v_i$\ubc88 \uc815\uc810\uc744 \uc5f0\uacb0\ud558\ub294 \uac00\uc911\uce58 $w_i$\uc758 \uac04\uc120\uc774 \uc874\uc7ac\ud55c\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4. $(1\\le u_i, v_i\\le V;$ $u_i \\ne v_i;$ $1\\le w_i\\le 10^6)$<\/p>\r\n\r\n<p>\uc8fc\uc5b4\uc9c0\ub294 \uadf8\ub798\ud504\ub294 \uc5f0\uacb0 \uadf8\ub798\ud504\uc774\uba70, \uc784\uc758\uc758 \ub450 \uc815\uc810 \uc0ac\uc774\uc758 \uac04\uc120\uc740 \ucd5c\ub300 \ud55c \uac1c \uc874\uc7ac\ud568\uc774 \ubcf4\uc7a5\ub41c\ub2e4.<\/p>\r\n","output":"<p>$G$\uc5d0 \ub300\ud574 &ldquo;\ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac \ub2e4\uc2dc \uadf8\ub9ac\uae30 \ub180\uc774&rdquo;\ub97c \ud1b5\ud574 \uadf8\ub9b4 \uc218 \uc788\ub294 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\uc758 \uae30\ub313\uac12\uc744 $10^9+7$\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uae30\uc57d \ubd84\uc218 $\\frac{p}{q}(p\\ge 0,q&gt;0,\\gcd(p,q) =1)$\ub97c $M$\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub294 $q^{-1}$\uac00 $q\\cdot q^{-1}\\equiv 1\\pmod M$\uc744 \ub9cc\uc871\ud558\ub294 \uc815\uc218, \uc989 $q$\uc758 $M$\uc5d0 \ub300\ud55c \ubaa8\ub4c8\ub85c \uacf1\uc148 \uc5ed\uc6d0\uc77c \ub54c, $p\\cdot q^{-1}\\pmod M$\ub85c \uc815\uc758\ud55c\ub2e4. \ub9cc\uc57d \uc815\uc218\uc77c \uacbd\uc6b0 $q=q^{-1}=1$\uc774\ubbc0\ub85c $p\\pmod M$\ub97c \uc758\ubbf8\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ub9cc\uc57d \uc815\ub2f5\uc758 \ubd84\ubaa8\uac00 $10^9+7$\uc758 \ubc30\uc218\uc778 \uacbd\uc6b0 \uae30\ub313\uac12 \ub300\uc2e0 <span style=\"color:#e74c3c;\"><code>-1<\/code><\/span>\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","subtask1":"<p>$E = V-1$<\/p>\r\n","subtask2":"<p>$V\\le 4$<\/p>\r\n","subtask3":"<p>\ubaa8\ub4e0 $w_i = 1;$ $E=\\frac{V(V-1)}{2}$<\/p>\r\n","subtask4":"<p>\ubaa8\ub4e0 $w_i = 1$<\/p>\r\n","subtask5":"<p>\ubaa8\ub4e0 $w_i \\le 2$<\/p>\r\n","subtask6":"<p>\ucd94\uac00\uc801\uc778 \uc81c\ud55c \uc870\uac74 \uc5c6\uc74c<\/p>\r\n","sample_explain_1":"<p>\uc8fc\uc5b4\uc9c4 \uadf8\ub798\ud504\ub294 \uc544\ub798 \uadf8\ub9bc\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/f222f3b0-679a-4de5-a97f-af708aba62e3\/-\/preview\/\" style=\"height: 104px; width: 100px;\" \/><\/p>\r\n\r\n<p>\uc774 \uadf8\ub798\ud504\uc758 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\ub294 \ub2e4\uc74c $16$\uac00\uc9c0\ub85c, $\\frac{1}{16}$\uc758 \ud655\ub960\ub85c \ud558\ub098\ub97c \ud6c4\uc784\uc774 \uace0\ub978\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/50a041a9-6599-4dd5-8a89-59464d279c69\/-\/preview\/\" style=\"height: 405px; width: 400px;\" \/><\/p>\r\n\r\n<p>\uc5b4\ub5a4 \uadf8\ub798\ud504\ub97c \uace8\ub77c\ub3c4, $T^\\prime$\uc740 \ubc18\ub4dc\uc2dc \uc544\ub798\uc640 \uac19\uc740 \uc0c1\ud0dc\uac00 \ub418\uace0, \ub2e4\uc2dc $16$\uac1c\uc758 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\ub97c \ubaa8\ub450 \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/1fcbe82e-a2f2-474d-8904-95e876bb707f\/-\/preview\/\" style=\"height: 104px; width: 100px;\" \/><\/p>\r\n","sample_explain_2":"<p>\uc8fc\uc5b4\uc9c4 \uadf8\ub798\ud504\ub294 \uc544\ub798 \uadf8\ub9bc\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/7d91aac3-bfe3-4be8-8c83-0007a7588db5\/-\/preview\/\" style=\"height: 100px; width: 100px;\" \/><\/p>\r\n\r\n<p>\uc774 \uadf8\ub798\ud504\uc758 \ucd5c\uc18c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\ub294 \ub2e4\uc74c $2$\uac00\uc9c0\ub85c, $\\frac{1}{2}$\uc758 \ud655\ub960\ub85c \ud558\ub098\ub97c \ud6c4\uc784\uc774 \uace0\ub978\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/03c9beac-05f1-4c87-9865-8664f6e548ca\/-\/preview\/\" style=\"width: 400px; height: 126px;\" \/><\/p>\r\n\r\n<p>\uc774\ud6c4 \uac00\ub2a5\ud55c \uc870\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. \uae30\ub313\uac12\uc740 $\\frac{3}{2}$\uc774\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/5119e4fb-6228-4b2b-a161-cc1393ed1e84\/-\/preview\/\" style=\"height: 432px; width: 400px;\" \/><\/p>\r\n"},{"problem_id":"31545","problem_lang":"1","title":"The MST Redrawing Game","description":"<p>Hanul recently joined the army. Upon learning that Hanul had been coding, a senior member of the army taught him the following &ldquo;MST Redrawing Game&rdquo;.<\/p>\r\n\r\n<ul>\r\n\t<li>The senior member first gives the junior member an undirected connected graph $G$. The weights of the edges of $G$ are all positive integers of $10^6$ or less.<\/li>\r\n\t<li>The junior member chooses a minimum spanning tree $T$ with the same probability among all different minimum spanning trees that can be drawn from $G$.<\/li>\r\n\t<li>The senior member tells the junior member an integer $x$, in order from $1$ to $10^6$, with an independent probability of $\\frac{1}{x}$. Since the senior member always tells the number $1$, the total number of numbers he tells is not less than $1$ and not more than&nbsp;$10^6$.<\/li>\r\n\t<li>When the senior member tells the&nbsp;integer $x$, the junior member erases <strong>all<\/strong> edges with a weight of $x$ on $T$. Let $T^\\prime$ be the state in which the edges are erased from $T$.<\/li>\r\n\t<li>The junior member selects&nbsp;$0$ or more&nbsp;edges from the original graph $G$ and adds them to $T^\\prime$ to create a minimum spanning tree of $G$.<\/li>\r\n<\/ul>\r\n\r\n<p>Hanul wondered how many minimum spanning trees he could draw in this game. Given the graph $G$ by the senior member, find the expected value of the number of minimum spanning trees that can be drawn in the MST Redrawing Game.<\/p>\r\n","input":"<p>The first line contains two integers $V$ and $E$, separated by spaces $(2\\le V\\le 500;$ $V-1\\le E\\le\\frac{V(V-1)}{2} )$ &ndash; the number of vertices and edges of $G$.<\/p>\r\n\r\n<p>The next $E$ lines each contain three integers $u_i$, $v_i$, and $w_i$, separated by spaces $(1\\le u_i,v_i\\le V;$ $u_i\\ne v_i;$ $1\\le w_i\\le 10^6)$ &ndash; indicating that there is an edge of $G$ with a weight of $w_i$ that connects two vertices $u_i$ and $v_i$.<\/p>\r\n\r\n<p>The given graph is a connected graph, and it is guaranteed that at most one edge exists between any two vertices.<\/p>\r\n","output":"<p>Output a single integer &ndash; the remainder of the expected value of the number of minimum spanning trees that can be drawn in the MST Redrawing Game using the graph $G$, divided by $10^9+7$. Note that $10^9+7$ is a prime number.<\/p>\r\n\r\n<p>The remainder of an irreducible fraction $\\frac{p}{q}(p\\ge 0,q&gt;0,\\gcd(p,q) =1)$ divided by $M$ is defined as $p\\cdot q^{-1}\\pmod M$, where $q^{-1}$ is a modulo multiplicative inverse of $q$ with respect to the modulus $M$, namely $q\\cdot q^{-1}\\equiv 1\\pmod M$. If the fraction is an integer, the value is $p\\pmod M$, since $q=q^{-1}=1$.<\/p>\r\n\r\n<p>When the denominator of the irreducible fraction of the expected value is a multiple of $10^9+7$, output <span style=\"color:#e74c3c;\"><code>-1<\/code><\/span> instead.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","subtask1":"<p>$E = V-1$<\/p>\r\n","subtask2":"<p>$V\\le 4$<\/p>\r\n","subtask3":"<p>All $w_i = 1;$&nbsp;$E=\\frac{V(V-1)}{2}$<\/p>\r\n","subtask4":"<p>All $w_i = 1$<\/p>\r\n","subtask5":"<p>All $w_i \\le 2$<\/p>\r\n","subtask6":"<p>There are no additional restrictions.<\/p>\r\n","sample_explain_1":"<p>The given graph is as below.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/f222f3b0-679a-4de5-a97f-af708aba62e3\/-\/preview\/\" style=\"height: 104px; width: 100px;\" \/><\/p>\r\n\r\n<p>There is a total of $16$ minimum spanning trees, and one of them is chosen, each with a probability of $\\frac{1}{16}$.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/50a041a9-6599-4dd5-8a89-59464d279c69\/-\/preview\/\" style=\"height: 405px; width: 400px;\" \/><\/p>\r\n\r\n<p>In any case, the state of $T^\\prime$ is as below, and all $16$ minimum spanning trees can be redrawn.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/1fcbe82e-a2f2-474d-8904-95e876bb707f\/-\/preview\/\" style=\"height: 104px; width: 100px;\" \/><\/p>\r\n","sample_explain_2":"<p>The given graph is as below.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/7d91aac3-bfe3-4be8-8c83-0007a7588db5\/-\/preview\/\" style=\"height: 100px; width: 100px;\" \/><\/p>\r\n\r\n<p>There is a total of $2$ minimum spanning trees, and one of them is chosen, each with a probability of $\\frac{1}{2}$.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/03c9beac-05f1-4c87-9865-8664f6e548ca\/-\/preview\/\" style=\"width: 400px; height: 126px;\" \/><\/p>\r\n\r\n<p>The possible combinations for erasing the lines and redrawing the minimum spanning trees are as below. The expected value is $\\frac{3}{2}$.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/5119e4fb-6228-4b2b-a161-cc1393ed1e84\/-\/preview\/\" style=\"height: 432px; width: 400px;\" \/><\/p>\r\n"}]